Packing Edge Disjoint Triangles: A Parameterized View Luke Mathieson Elena Prieto Peter Shaw School of Electrical Engineering and Computer Science The University of Newcastle Australia Abstract. The problem of packing k edge-disjoint triangles in a graph has been thoroughly studied both in the classical complexity and the approximation fields and it has a wide range of applications in many areas, especially computational biology [BP96]. In this paper we present an analysis of the problem from a parameterized complexity viewpoint. We describe a fixed-parameter tractable algorithm for the problem by means of kernelization and crown rule reductions, two of the newest techniques for fixed-parameter algorithm design. We achieve a kernel size bounded by 4k, where k is the number of triangles in the packing. 1 Introduction The problem of finding the maximum number of vertex or edge disjoint cycles in an undirected graph G has applications in many different fields, for instance in computational biology [BP96]. The problem is defined as follows: Let G =(V,E) be a simple graph. A triangle T in G is any induced subgraph of G having precisely 3 edges and 3 vertices. A graph G =(V,E) is said to have a k packing of triangles if there exist k disjoint copies T 1 , ..., T k of T in the vertex set of G. The packing is called vertex-disjoint if T 1 , ..., T k share no vertices and is called edge-disjoint if we allow T 1 , ..., T k to have some vertices in common but no edges exist in T i ∩ T j when i = j . In this paper we look at the parameterized version of the edge-disjoint case. More formally, we will study in detail the parameterized complexity of the following problem: k-Edge Disjoint Triangle Packing (ETP) Instance: A graph G =(V,E), a positive integer k Parameter: k Question: Are there at least k edge disjoint instances of T in G? This problem is NP-hard for general graphs [HOL81] and has also been shown to be NP-hard for planar graphs, even if the maximum degree is 5 or more. Regarding approximability, ETP is known to be APX-hard [K94]. A general result of [HS89] leads to a polynomial time (3/2+ ǫ) approximation algorithm for any ǫ> 0 for this problem. If G is planar Baker [B94] gives a polynomial time approximation scheme for the vertex-disjoint case which can be extended to